Worksheet: Fundamental Theorem of Line Integrals

In this worksheet, we will practice determining whether a vector field is conservative by finding a suitable function which produces the vector field as its gradient.

Q1:

Is there a potential for ? If so, find one.

• A yes,
• B no
• C yes,
• D yes,
• E yes,

Q2:

Is there a potential for ? If so, find one.

• A yes,
• B no
• C yes,
• D yes,
• E yes,

Q3:

Is there a potential for ? If so, find one.

• A yes,
• B no
• C yes,
• D yes,
• E yes,

Q4:

Is there a potential for ? If so, find one.

• A no
• B yes,

Q5:

Is there a potential for ? If so, find one.

• A yes,
• B yes,
• C yes,
• D no
• E yes,

Q6:

State whether or not the vector field where , , are constants has a potential in .

• Ayes
• Bno

Q7:

State whether or not the vector field has a potential in .

• Ano
• Byes

Q8:

Is there a potential for ? If so, find one.

• A yes,
• B yes,
• C yes,
• Dno
• E yes,

Q9:

Which of the following is true concerning the vector field defined by ?

• A has a positive divergence at every point .
• B , for all .
• C is a conservative vector field.
• D is orthogonal to the vector , for all .